Coupon coloring of cographs

Coupon coloring is a new coloring which has many applications. A k-coupon coloring of a graph G is a k-coloring of G by colors [k]={1,2,…,k} such that the neighborhood of every vertex of G contains vertices of all colors from [k]. The maximum integer k for which a k-coupon coloring exists is called the coupon coloring number of G, and it is denoted by χc(G). In this paper, we studied the coupon coloring of cographs, which are graphs that can be generated from the single vertex graph K1 by complementation and disjoint union, and have applications in many interesting problems. We use the cotree representation of a cograph to give a polynomial time algorithm to color the vertices of a cograph, and then prove that this coloring is a coupon coloring with maximum colors, hence get the coupon coloring numbers of the cograph.